the likelihood
The likelihood measures how well a hypothesis explains the evidence you actually observed. In Bayes' theorem it is the P(B given A) term: if the hypothesis A were true, how probable would the observed evidence B be? A test that almost always reads positive when the disease is present has a high likelihood for the positive result; a test that rarely does has a low one. The likelihood is the channel through which the data speaks.
There is a subtle but important point of language. P(B given A) is read in two directions. As a function of B (the data) with the hypothesis A fixed, it is an ordinary conditional probability and sums to 1 over all possible data. But as a function of A (the hypothesis) with the data B fixed at what you saw, it is called the likelihood — and as a function of the hypothesis it need NOT sum to 1; it is not a probability distribution over hypotheses. We compare likelihoods across hypotheses to see which one makes the observed data least surprising.
Likelihood is the engine of evidence. Bayes says posterior is proportional to prior times likelihood, so the likelihood is exactly the factor that reweights your prior beliefs in light of what happened. The ratio of likelihoods under two competing hypotheses — the likelihood ratio — is the cleanest single number summarizing how strongly the evidence favors one hypothesis over the other, and it drives the odds form of Bayes.
You see 8 heads in 10 tosses. Under 'fair coin' the likelihood of that exact data is C(10,8)(0.5)^10 = about 0.044. Under 'biased, p = 0.8' it is C(10,8)(0.8)^8(0.2)^2 = about 0.302. The biased hypothesis makes the data far more likely — its likelihood is roughly 7 times higher.
Likelihood asks: how well does each hypothesis predict the data we actually got?
A high likelihood does not by itself make a hypothesis probable — you must still multiply by its prior. A bizarre hypothesis can fit data perfectly yet stay improbable because its prior is tiny.